Unveiling Quadratic Secrets: Analyzing Q(x) = X² + 9x + 17

Hey guys! Let's dive into a classic quadratic function and break down its secrets. We'll be working with the function q(x) = x² + 9x + 17. Our goal? To explore its vertex, intercepts, and overall shape. It's like a fun puzzle where we get to understand how these equations work and how to visualize them. So, grab your pens and paper (or your favorite graphing tool!), and let's get started. We'll be covering a few key aspects, making sure we have a solid understanding of this quadratic function. We'll rewrite the function, pinpoint its most important part (the vertex), find where it crosses the axes, draw it on a graph, and, finally, figure out its line of symmetry.

(a) Rewriting q(x) in Vertex Form: Completing the Square

Alright, first things first: we need to rewrite q(x) in vertex form. Why? Because the vertex form, f(x) = a(x - h)² + k, is like the function's secret code that tells us the exact location of the vertex, which is super important for understanding the function's behavior. To do this, we'll use a technique called 'completing the square.' Don't worry, it sounds scarier than it is! The goal is to manipulate our original equation into that nice, clean vertex form. So we will make it happen, by following these simple steps.

We begin with our given function: q(x) = x² + 9x + 17. Notice the coefficient of our x² term is 1, which makes our lives easier. If it weren't, we'd have to factor it out first. Now, we focus on the x terms, x² + 9x. To complete the square, we take half of the coefficient of our x term (which is 9), square it, and then add and subtract that value within the equation. Half of 9 is 4.5, and 4.5 squared is 20.25.

Here’s how it looks:

q(x) = (x² + 9x + 20.25) + 17 - 20.25

See what we did there? We added and subtracted 20.25 to keep the equation balanced. Now, the expression in the parentheses, (x² + 9x + 20.25), is a perfect square trinomial! We can rewrite it as (x + 4.5)². And that is what we are looking for. Now, let’s simplify the constant terms: 17 - 20.25 = -3.25. This leaves us with our final vertex form equation:

q(x) = (x + 4.5)² - 3.25

Boom! We've successfully rewritten the function in vertex form. We can see how this form is useful because it is easy to find the vertex coordinates. It is like unlocking the first level of a video game. We've now got the function in the form f(x) = a(x - h)² + k. Now it’s just a matter of identifying the values.

(b) Identifying the Vertex: The Function's Highest or Lowest Point

Now that we have the function in vertex form, identifying the vertex is a piece of cake. The vertex form f(x) = a(x - h)² + k gives us the coordinates of the vertex directly. The vertex is the point (h, k). In our equation, q(x) = (x + 4.5)² - 3.25, we can see that a = 1, h = -4.5, and k = -3.25. Remember that the h value is the opposite sign in the equation. So, if we have (x + 4.5), it means h is -4.5. If the equation was (x - 4.5) then h would be 4.5. It's really that simple.

Therefore, the vertex of the parabola is at the point (-4.5, -3.25). This point is crucial because it represents either the minimum (if the parabola opens upwards, as it does here because a=1, and 1>0) or the maximum (if the parabola opens downwards, if a<0) value of the function. In our case, since the coefficient a is positive, the parabola opens upwards, and the vertex is the minimum point. This means that the lowest point on the graph is at (-4.5, -3.25). If we were to use a graphing calculator, we would have similar results. The vertex helps us understand the function's range and its overall shape. We are basically getting closer to being experts at understanding quadratics.

(c) Determining the x-intercept(s): Where the Function Crosses the x-axis

Next up, we need to find the x-intercepts, or the points where the graph of the function crosses the x-axis. At the x-intercept(s), the value of y (or q(x)) is equal to zero. Therefore, we will be solving the equation q(x) = 0. So, let’s set our equation to zero and solve it for x. We can do this using the original equation or the vertex form, but the quadratic formula is usually the easiest route. We'll use the original equation q(x) = x² + 9x + 17 = 0. The quadratic formula is:

x = (-b ± √(b² - 4ac)) / 2a

Where a, b, and c are the coefficients from our quadratic equation. In this case, a = 1, b = 9, and c = 17. Now we plug in the numbers:

x = (-9 ± √(9² - 4 * 1 * 17)) / (2 * 1)

Simplify the formula and we get:

x = (-9 ± √(81 - 68)) / 2

x = (-9 ± √13) / 2

Now, we have two possible solutions, or x-intercepts:

x = (-9 + √13) / 2 ≈ -2.697

x = (-9 - √13) / 2 ≈ -6.303

So, the x-intercepts are approximately (-2.697, 0) and (-6.303, 0). These are the points where the parabola crosses the x-axis. Knowing the x-intercepts is helpful because it gives you another way to sketch the graph and understand its position relative to the x-axis. Using a graphing calculator could confirm these results. This is similar to plotting a coordinate on a map; it indicates the specific points where the function touches the x-axis. Therefore, we know where to plot them to create the shape of the graph.

(d) Determining the y-intercept(s): Where the Function Crosses the y-axis

Finding the y-intercept is usually a pretty straightforward task. The y-intercept is the point where the graph crosses the y-axis. At this point, the value of x is equal to zero. All we have to do is plug in x = 0 into our original equation and solve for q(x).

q(x) = x² + 9x + 17

q(0) = 0² + 9 * 0 + 17

q(0) = 17

This means that the y-intercept is at the point (0, 17). This is where the parabola intersects the y-axis. The y-intercept is the point where the graph 'starts' on the y-axis. Again, this helps us to visualize the graph. It also helps to see how the function is translated up or down compared to the basic parabola. This point gives us another point to help us sketch the function accurately. It completes one more of the most important points of the function.

(e) Sketching the Function: Putting It All Together

Alright, it's time to bring everything we've found together and sketch the function. We have all the important pieces: the vertex, the x-intercepts, and the y-intercept. Let's make a mental checklist before we begin:

1. Vertex: (-4.5, -3.25) - This is our minimum point, the bottom of the U-shape.
2. x-intercepts: (-2.697, 0) and (-6.303, 0) - These are the points where the parabola crosses the x-axis.
3. y-intercept: (0, 17) - This is where the parabola crosses the y-axis.

Knowing a = 1, the parabola opens upwards. Start by plotting the vertex, then plot the x-intercepts and y-intercept. Sketch a smooth, U-shaped curve that passes through these points. Remember that the parabola is symmetrical, so the two 'arms' of the U should be roughly the same distance from the axis of symmetry. You can also use additional points to get a better sketch.

Another way you can sketch the function, is by using a table of values and plotting the points. For example, choose some x-values around the vertex (like -6, -5, -4, -3, -2, -1, 0, 1, 2) and calculate the corresponding q(x) values using the equation q(x) = x² + 9x + 17. Plot those points and connect them to form the parabola. This method ensures that the graph is accurately drawn and represents the function.

Sketching the function is essential because it is a visual representation of the function's behavior. It allows us to see how the changes in x affect the value of q(x). It's a way of making the abstract concepts of algebra feel more concrete.

(f) Determining the Axis of Symmetry: The Parabola's Mirror

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two symmetrical halves. It's like the mirror line of the U-shape. Finding the axis of symmetry is super easy once you know the vertex. The equation of the axis of symmetry is x = h, where h is the x-coordinate of the vertex. In our case, the vertex is (-4.5, -3.25), so the axis of symmetry is the vertical line x = -4.5. The axis of symmetry helps us understand the symmetry of the parabola. It ensures that the two sides of the curve are mirror images of each other. Using this line as a reference helps with graphing. It is another important aspect when plotting the function and knowing the x-intercepts and the vertex makes everything easier and more accurate.

And that's it, guys! We've successfully analyzed the quadratic function q(x) = x² + 9x + 17. We rewrote it in vertex form, identified the vertex, found the intercepts, sketched the function, and determined its axis of symmetry. You're now a little more familiar with quadratics and how to work with them. Keep practicing, and you'll become a quadratic master in no time!